Math and Physics for Everyone

arithmetic geometry

In many posts on this blog, such as Basics of Arithmetic Geometry and Elliptic Curves, we have discussed how the geometry of shapes described by polynomial equations is closely related to number theory. This is especially true when it comes to the thousands-of-years-old subject of Diophantine equations, polynomial equations whose coefficients are whole numbers, and whose solutions of interest are also whole numbers (or, equivalently, rational numbers, since we can multiply or divide both sides of the polynomial equation by a whole number). We might therefore expect that the more modern and more sophisticated tools of algebraic geometry (which is a subject that started out as just the geometry of shapes described by polynomial equations) might be extremely useful in answering questions and problems in number theory.

One of the tools we can use for this purpose is the concept of an arithmetic scheme, which makes use of the ideas we discussed in Grothendieck’s Relative Point of View. An arithmetic variety is defined to be a a regular scheme that is projective and flat over the scheme . An example of this is the scheme , which is two-dimensional, and hence also referred to as an arithmetic surface.

We recall that the points of an affine scheme , for some ring , are given by the prime ideals of . Therefore the scheme has one point for every prime ideal – one “closed point” for every prime number , and a “generic point” given by the prime ideal .

However, we also recall from Adeles and Ideles the concept of the “infinite primes” – which correspond to the archimedean valuations of a number field, just as the finite primes (primes in the classical sense) correspond to the nonarchimedean valuations. It is important to consider the infinite primes when dealing with questions and problems in number theory, and therefore we need to modify some aspects of algebraic geometry if we are to use it in helping us with number theory.

We now go back to arithmetic schemes, taking into consideration the infinite prime. Since we are dealing with the ordinary integers , there is only one infinite prime, corresponding to the embedding of the rational numbers into the real numbers. More generally we can also consider an arithmetic variety over instead of , where is the ring of integers of a number field . In this case we may have several infinite primes, corresponding to the embediings of into the real and complex numbers. In this post, however, we will consider only and one infinite prime.

How do we describe an arithmetic scheme when the scheme has been “compactified” with the infinite prime? Let us look at the fibers of the arithmetic scheme. The fiber of an arithmetic scheme at a finite prime is given by the scheme defined by the same homogeneous polynomials as , but with the coefficients taken modulo , so that they are elements of the finite field . The fiber over the generic point is given by taking the tensor product of the coordinate ring of with the rational numbers. But how should we describe the fiber over the infinite prime?

It was the idea of the mathematician Suren Arakelov that the fiber over the infinite prime should be given by a complex variety – in the case of an arithmetic surface, which Arakelov originally considered, the fiber should be given by a Riemann surface. The ultimate goal of all this machinery, at least when Arakelov was constructing it, was to prove the famous Mordell conjecture, which states that the number of rational solutions to a curve of genus greater than or equal to was finite. These rational solutions correspond to sections of the arithmetic surface, and Arakelov’s strategy was to “bound” the number of these solutions by constructing a “height function” using intersection theory (see Algebraic Cycles and Intersection Theory) on the arithmetic surface. Arakelov unfortunately was not able to carry out his proof. He had only a very short career, being forced to retire after being diagnosed with schizophrenia. The Mordell conjecture was eventually proved by another mathematician, Gerd Faltings, who continues to develop Arakelov’s ideas.

We say that is the vector space of complex-valued differential forms of type . We have differential operators

.

.

We let be the dual to the vector space , and we write to denote . We refer to an element of as a current of type . We have an inclusion map

mapping a differential form of type to a current of type , given by

for all .

The differential operators , , , and induce maps , , and on . We define the maps , , and on by

We also define

.

For every irreducible analytic subvariety of codimension , we define the current by

for all , where is the nonsingular locus of .

A Green current for a codimension analytic subvariety is defined to be an element of such that

for some .

Let be the resolution of singularities of . This means that there exists a proper map such that is smooth, is a divisor with normal crossings (this means that each irreducible component of is nonsingular, and whenever they meet at a point their local equations are linearly independent) whenever contains the singular locus of , and is an isomorphism.

A smooth form on is said to be of logarithmic type along if there exists a projective map such that is a divisor with normal crossings, is smooth, and is the direct image by of a form on satisfying the following equation

where is a local equation of for every in , are and closed smooth forms, and is a smooth form.

For every irreducible subvariety there exists a smooth form on of logarithmic type along such that is a Green current for :

where w is smooth on X. We say that is a Green current of logarithmic type.

We now proceed to discuss this intersection theory on the arithmetic scheme. We consider a vector bundle on the arithmetic scheme , a holomorphic vector bundle (a complex vector bundle such that the projection map is holomorphic) on the fibers at the infinite prime, and a smooth hermitian metric (a sesquilinear form with the property that ) on which is invariant under the complex conjugation on . We refer to this collection of data as a hermitian vector bundle on .

Given an arithmetic scheme and a hermitian vector bundle on , we can define associated “arithmetic”, or “Arakelov-theoretic” (i.e. taking into account the infinite prime) analogues of the algebraic cycles and Chow groups that we discussed in Algebraic Cycles and Intersection Theory.

An arithmetic cycle on is a pair where is an algebraic cycle on , i.e. a linear combination of closed irreducible subschemes of , of some fixed codimension , with integer coefficients , and is a Green current for , i.e. satisfies the equation

where

for differential forms and of appropriate degree.

We define the arithmetic Chow group as the group of arithmetic cycles modulo the subgroup generated by the pairs and , where and are currents of appropriate degree and is some rational function on some irreducible closed subscheme of codimension in .

Next we want to have an intersection product on Chow groups, i.e. a bilinear pairing

We now define this intersection product. Let and . Assume that and are irreducible. Let , and . If and intersect properly, i.e. , then we have

where is just the usual intersection product of algebraic cycles, and is the -product of Green currents, defined for a Green current of logarithmic type and a Green current , where and are closed irreducible subsets of with not contained in , as

where

and

for is the resolution of singularities of composed with the inclusion of into .

In the case that and do not intersect properly, there is a rational function on such that and intersect properly, and if is another rational function such that and intersect properly, the cycle

is in the subgroup . Here the notation refers to the pair .

This concludes our little introduction to arithmetic intersection theory. We now give a short discussion what else can be done with such a theory. The mathematicians Henri Gillet and Christophe Soule used this arithmetic intersection theory to construct arithmetic analogues of Chern classes, Chern characters, Todd classes, and the Grothendieck-Riemann-Roch theorem (see Chern Classes and Generalized Riemann-Roch Theorems). These constructions are not so straightforward – for instance, one has to deal with the fact that unlike the classical case, the arithmetic Chern character is not additive on exact sequences. This failure to be additive on exact sequences is measured by the Bott-Chern character. The Bott-Chern character plays a part in defining the arithmetic analogue of the Grothendieck group .

In order to define the arithmetic analogue of the Grothendieck-Riemann-Roch theorem, one must then define the direct image map for a proper flat map of arithmetic varieties. This involves constructing a canonical line bundle on , whose fiber at is the determinant of cohomology of , i.e.

as well as a metric , called the Quillen metric, on . With such a direct image map we can now give the statement of the arithmetic Grothendieck-Riemann-Roch theorem. It was originally stated by Gillet and Soule in terms of components of degree one in the arithmetic Chow group :

where denotes the arithmetic Chern character, denotes the arithmetic Todd class, is the relative tangent bundle of , is the map from

to sending the element in to the class of in , and

.

Later on Gillet and Soule formulated the arithmetic Grothendieck-Riemann-Roch theorem in higher degree as

for .

Aside from the work of Gillet and Soule, there is also the work of the mathematician Amaury Thuillier making use of ideas from -adic geometry, constructing a nonarchimedean potential theory on curves that allows the finite primes and the infinite primes to be treated on a more equal footing, at least for arithmetic surfaces. The work of Thuillier is part of ongoing efforts to construct an adelic geometry, which is hoped to be the next stage in the evolution of Arakelov geometry.

A few days ago, from July 17 to 25, I attended the SEAMS (Southeast Asian Mathematical Society) School held at the Institute of Mathematics, University of the Philippines Diliman, discussing topics on elliptic curves. The school was also partially supported by CIMPA (Centre International de Mathematiques Pures et Appliquees, or International Center for Pure and Applied Mathematics), and I believe also by the Roman Number Theory Association and the Number Theory Foundation. Here’s the official website for the event:

There were many participants from countries all over Southeast Asia, including Indonesia, Malaysia, Philippines, and Vietnam, as well as one participant from Austria and another from India. The lecturers came from Canada, France, Italy, and Philippines.

Jerome Dimabayao and Michel Waldschmidt started off the school, introducing the algebraic and analytic aspects of elliptic curves, respectively. We have tackled these subjects in this blog, in Elliptic Curves and The Moduli Space of Elliptic Curves, but the school discussed them in much more detail; for instance, we got a glimpse of how Karl Weierstrass might have come up with the function named after him, which relates the equation defining an elliptic curve to a lattice in the complex plane. This requires some complex analysis, which unfortunately we have not discussed that much in this blog yet.

Francesco Pappalardi then discussed some important theorems regarding rational points on elliptic curves, such as the Nagell-Lutz theorem and the famous Mordell-Weil theorem. Then, Julius Basilla discussed the counting of points of elliptic curves over finite fields, often making use of the Hasse-Weil inequality which we have discussed inThe Riemann Hypothesis for Curves over Finite Fields, and the applications of this theory to cryptography. Claude Levesque then introduced to us the fascinating theory of quadratic forms, which can be used to calculate the class number of a quadratic number field (see Algebraic Numbers), and the relation of this theory to elliptic curves.

Richell Celeste discussed the reduction of elliptic curves modulo primes, a subject which we have also discussed here in the post Reduction of Elliptic Curves Modulo Primes, and two famous problems related to elliptic curves, Fermat’s Last Theorem, which was solved by Andrew Wiles in 1995, and the still unsolved Birch and Swinnerton-Dyer conjecture regarding the rank of the group of rational points of elliptic curves. Fidel Nemenzo then discussed the classical problem of finding “congruent numbers“, rational numbers forming the sides of a right triangle whose area is given by an integer, and the rather surprising connection of this problem to elliptic curves.

On the last day of the school, Jerome Dimabayao discussed the fascinating connection between elliptic curves and Galois representations, which we have given a passing mention to at the end of the post Elliptic Curves. Finally, Jared Guissmo Asuncion gave a tutorial on the software PARI which we can use to make calculations related to elliptic curves.

Participants were also given the opportunity to present their research work or topics they were interested in. I gave a short presentation discussing certain aspects of algebraic geometry related to number theory, focusing on the spectrum of the integers, and a mention of related modern mathematical research, such as Arakelov theory, and the view of the integers as a curve (under the Zariski topology) and as a three-dimensional manifold (under the etale topology).

Aside from the lectures, we also had an excursion to the mountainous province of Rizal, which is a short distance away from Manila, but provides a nice getaway from the environment of the big city. We visited a couple of art museums (one of which was also a restaurant serving traditional Filipino cuisine), an underground cave system, and a waterfall. We used this time to relax and talk with each other, for instance about our cultures, and many other things. Of course we still talked about mathematics, and during this trip I learned about many interesting things from my fellow participants, such as the class field theory problem and the subject of real algebraic geometry .

I believe lecture notes will be put up on the school website at some point by some of the participants of the school. For now, some of the lecturers have put up useful references for their lectures.

SEAMS School Manila 2017 was actually the first summer school or conference of its kind that I attended in mathematics, and I enjoyed very much the time I spent there, not only in learning about elliptic curves but also making new friends among the mathematicians in attendance. At some point I also hope to make some posts on this blog regarding the interesting things I have learned at that school.

We have discussed elliptic curves over the rational numbers, the real numbers, and the complex numbers in Elliptic Curves. In this post, we discuss elliptic curves over finite fields of the form , where is a prime, obtained by “reducing” an elliptic curve over the integers modulo (see Modular Arithmetic and Quotient Sets).

We recall that in Elliptic Curves we gave the definition of an elliptic curve as a polynomial equation that we may write as

with and satisfying the condition that

.

Still, we claimed that we will not be able to write the equation of the elliptic curve when the coefficients of the elliptic curve are of characteristic equal to or , as is the case for the finite fields or , therefore we will give more general forms for the equation of the elliptic curve later, along with the appropriate conditions. To help us with the latter, we will first look at the case of curves over the real numbers, where we can still make use of the equations above, and see what happens when the conditions on and are not satisfied.

Let both and both be equal to , in which case the condition is not satisfied. Then our curve (which is not an elliptic curve) is given by the equation

whose graph in the – plane is given by the following figure (plotted using the WolframAlpha software):

Next let and . Once again the condition is not satisfied. Our curve is given by

and whose graph is given by the following figure (again plotted using WolframAlpha):

Note also that in both cases, the right hand side of the equations of the curves are polynomials in with a double or triple root; for , the right hand side, , has a triple root at , while for , the right hand side, , factors into and therefore has a double root at .

The two curves, and , are examples of singular curves. It is therefore a requirement for a curve to be an elliptic curve, that it must be nonsingular.

We now introduce the general form of an elliptic curve, applicable even when the coefficients belong to fields of characteristic or , along with the general condition for it to be nonsingular. We note that the elliptic curve has a “point at infinity“; in order to make this idea explicit, we make use of the notion of projective space (see Projective Geometry) and write our equation in homogeneous coordinates, , and :

This equation is called the long Weierstrass equation. We may also say that it is in long Weierstrass form.

We can now define what it means for a curve to be singular. Let

Then a singular point on this curve is a point with coordinates , , and such that

It might be difficult to think of calculus when we are considering, for example, curves over finite fields, where there are a finite number of points on the curve, so we might instead just think of the partial derivatives of the curve as being obtained “algebraically” using the “power rule” of basic calculus,

and applying it, along with the usual rules for partial derivatives and constant factors, to every term of the curve. Such is the power of algebraic geometry; it allows us to “import” techniques from calculus and other areas of mathematics which we would not ordinarily think of as being applicable to cases such as curves over finite fields.

If a curve has no singular points, then it is called a nonsingular curve. We may also say that the curve is smooth. In order for a curve written in long Weierstrass form to be an elliptic curve, we require that it be a nonsingular curve as well.

If the coefficients of the curve are not of characteristic equal to , we can make a projective transformation of variables to write its equation in a simpler form, known as the short Weierstrass equation, or short Weierstrass form:

In this case the condition for the curve to be nonsingular can be written in the following form:

The quantity

is called the discriminant of the curve.

We note now, of course, that the usual expressions for the elliptic curve, in what we call affine coordinates and , can be recovered from our expression in terms of homogeneous coordinates, , and simply by setting and . The case of course corresponds to the “point at infinity”.

We now consider an elliptic curve whose equation has coefficients which are rational numbers. We can make a projective transformation of variables to rewrite the equation into one which has integers as coefficients. Then we can reduce the coefficients modulo a prime and investigate the points of the elliptic curve considered as having coordinates in the finite field .

It may happen that when we reduce an elliptic curve modulo , the resulting curve over the finite field is no longer nonsingular. In this case we say that it has bad reduction at . Consider, for example, the following elliptic curve (written in affine coordinates):

Let us reduce this modulo the prime . Then, since and , we obtain the curve

over . The right hand side actually factors into over , which means that it has a double root at (which is equivalent to modulo ), and has discriminant equal to zero over , hence, this curve over is singular, and the elliptic curve given by has bad reduction at . It also has bad reduction at ; in fact, we mentioned earlier that we cannot even write an elliptic curve in the form when the field of coefficients have characteristic equal to . This is because such a curve will always be singular over such a field. The curve remains nonsingular over all other primes, however; we also say that the curve has good reduction over all primes except for and .

In the case that an elliptic curve has bad reduction at , we say that it has additive reduction if there is only one tangent line at the singular point (we also say that the singular point is a cusp), for example in the case of the curve , and we say that it has multiplicative reduction if there are two distinct tangent lines at the singular point (in this case we say that the singular point is a node), for example in the case of the curve . If the slope of these tangent lines are given by elements of the same field as the coefficients of the curve (in our case rational numbers), we say that it has split multiplicative reduction, otherwise, we say that it has nonsplit multiplicative reduction. We note that since we are working with finite fields, what we describe as “tangent lines” are objects that we must define “algebraically”, as we have done earlier when describing the notion of a curve being singular.

As we have already seen in The Riemann Hypothesis for Curves over Finite Fields, whenever we have a curve over some finite field (where for some natural number ), our curve will also have a finite number of points, and these points will have coordinates in . We denote the number of these points by . In our case, we are interested in the case , so that . When our elliptic curve has good reduction over , we define a quantity , sometimes called the -defect, or also known as the trace of Frobenius, as

.

We can now define the Hasse-Weil L-function of an elliptic curve as follows:

where runs over all prime numbers, and

if has good reduction at

if has split multiplicative reduction at

if has nonsplit multiplicative reduction at

if has additive reduction at .

The Hasse-Weil L-function encodes number-theoretic information related to the elliptic curve, and much of modern mathematical research involves this function. For example, the Birch and Swinnerton-Dyer conjecture says that the rank of the group formed by the rational points of the elliptic curve (see Elliptic Curves), also known as the Mordell-Weil group, is equal to the order of the zero of the Hasse-Weil L-function at , i.e. we have the following Taylor series expansion of the Hasse-Weil L-function at :

where is a constant and is the rank of the elliptic curve.

Meanwhile, the Shimura-Taniyama-Weil conjecture, now also known as the modularity conjecture, central to Andrew Wiles’s proof of Fermat’s Last Theorem, states that the Hasse-Weil L-function can be expressed as the following series:

For more on the modularity theorem and Wiles’s proof of Fermat’s Last Theorem, the reader is encouraged to read the award-winning article A Marvelous Proof by Fernando Q. Gouvea, which is freely and legally available online. A link to this article (hosted on the website of the Mathematical Association of America) is provided among the list of references below.

A moduli space is a kind of “parameter space” that “classifies” mathematical objects. Every point of the moduli space stands for a mathematical object, in such a way that mathematical objects which are more similar to each other are closer and those that are more different from each other are farther apart. We may use the notion of equivalence relations (see Modular Arithmetic and Quotient Sets) to assign several objects which are in some sense “isomorphic” to each other to a single point.

We have discussed on this blog before one example of a moduli space – the projective line (see Projective Geometry). Every point on the projective line corresponds to a geometric object, a line through the origin. Two lines which have almost the same value of the slope will be closer on the projective line compared to two lines which are almost perpendicular.

Another example of a moduli space is that for circles on a plane – such a circle is specified by three real numbers, two coordinates for the center and one positive real number for the radius. Therefore the moduli space for circles on a plane will consist of a “half-volume” of some sort, like 3D space except that one coordinate is restricted to be strictly positive. But if we only care about the circles up to “congruence”, we can ignore the coordinates for the center – or we can also think of it as simply sending circles with the same radius to a single point, even if they are centered at different points. This moduli space is just the positive real line. Every point on this moduli space, which is a positive real number, corresponds to all the circles with radius equal to that positive real number.

We now want to construct the moduli space of elliptic curves. In order to do this we will need to first understand the meaning of the following statement:

Over the complex numbers, an elliptic curve is a torus.

We have already seen in Elliptic Curves what an elliptic curve looks like when graphed in the – plane, where and are real numbers. This gives us a look at the points of the elliptic curve whose coordinates are real numbers, or to put it in another way, these are the real numbers and which satisfy the equation of the elliptic curve.

When we look at the points of the elliptic curve with complex coordinates, or in other words the complex numbers which satisfy the equation of the elliptic curve, the situation is more complicated. First off, what we actually have is not what we usually think of as a curve, but rather a surface, in the same way that the complex numbers do not form a line like the real numbers do, but instead form a plane. However, even though it is not easy to visualize, there is a function called the Weierstrass elliptic function which provides a correspondence between the (complex) points of an elliptic curve and the points in the “fundamental parallelogram” of a lattice in the complex plane. We can think of “gluing” the opposite sides of this fundamental parallelogram to obtain a torus. This is what we mean when we say that an elliptic curve is a torus. This also means that there is a correspondence between elliptic curves and lattices in the complex plane.

We will discuss more about lattices later on in this post, but first, just in case the preceding discussion seems a little contrived, we elaborate a bit on the Weierstrass elliptic function. We must first discuss the concept of a holomorphic function. We have discussed in An Intuitive Introduction to Calculus the concept of the derivative of a function. Now not all functions have derivatives that exist at all points; in the case that the derivative of the function does exist at all points, we refer to the function as a differentiable function.

The concept of a holomorphic function in complex analysis (analysis is the term usually used in modern mathematics to refer to calculus and its related subjects) is akin to the concept of a differentiable function in real analysis. The derivative is defined as the limit of a certain ratio as the numerator and the denominator both approach zero; on the real line, there are limited ways in which these quantities can approach zero, but on the complex plane, they can approach zero from several different directions; for a function to be holomorphic, the expression for its derivative must remain the same regardless of the direction by which we approach zero.

In previous posts on topology on this blog we have been treating two different topological spaces as essentially the same whenever we can find a bijective and continuous function (also known as a homeomorphism) between them; similarly, we have been treating different algebraic structures such as groups, rings, modules, and vector spaces as essentially the same whenever we can find a bijective homomorphism (an isomorphism) between two such structures. Following these ideas and applying them to complex analysis, we may treat two spaces as essentially the same if we can find a bijective holomorphic function between them.

The Weierstrass elliptic function is not quite holomorphic, but is meromorphic – this means that it would have been holomorphic everywhere if not for the “lattice points” where there exist “poles”. But it is alright for us, because such a lattice point is to be mapped to the “point at infinity”. All in all, this allows us to think of the complex points of the elliptic curve as being essentially the same as a torus, following the ideas discussed in the preceding paragraph.

Moreover, the torus has a group structure of its own, considered as the direct product group where is the group of complex numbers of magnitude equal to with the law of composition given by the multiplication of complex numbers. When the complex points of the elliptic curve get mapped by the Weierstrass elliptic function to the points of the torus, the group structure provided by the “tangent and chord” or “tangent and secant” construction becomes the group structure of the torus. In other words, the Weierstrass elliptic function provides us with a group isomorphism.

All this discussion means that the study of elliptic curves becomes the study of lattices in the complex plane. Therefore, what we want to construct is the moduli space of lattices in the complex plane, up to a certain equivalence relation – two lattices are to be considered equivalent if one can be obtained by multiplying the other by a complex number (this equivalence relation is called homothety). Going back to elliptic curves, this corresponds to an isomorphism of elliptic curves in the sense of algebraic geometry.

Now given two complex numbers and , a lattice in the complex plane is given by

For example, setting and , gives a “square” lattice. This lattice is also the set of all Gaussian integers. The fundamental parallelogram is the parallelogram formed by the vertices , , , and . Here is an example of a lattice, courtesy of used Alvaro Lozano Robledo of Wikipedia:

The fundamental parallelogram is in blue. Here is another, courtesy of user Sam Derbyshire of Wikipedia:

Because we only care about lattices up to homothety, we can “rescale” the lattice by multiplying it with a complex number equal to , so that we have a new lattice equivalent under homothety to the old one, given by

where

.

We can always interchange and , but we will fix our convention so that the complex number , when written in polar form always has a positive angle between 0 and 180 degrees. If we cannot obtain this using our choice of and , then we switch the two.

Now what this means is that a complex number , which we note is a complex number in the upper half plane , because of our convention in choosing and , uniquely specifies a homothety class of lattices . However, a homothety class of lattices may not always uniquely specify such a complex number . Several such complex numbers may refer to the same homothety class of lattices.

What and specify is a choice of basis (see More on Vector Spaces and Modules) for the lattice ; we may choose several different bases to refer to the same lattice. Hence, the upper half plane is not yet the moduli space of all lattices in the complex plane (up to homothety); instead it is an example of what is called a Teichmuller space. To obtain the moduli space from the Teichmuller space, we need to figure out when two different bases specify lattices that are homothetic.

We will just write down the answer here; two complex numbers and refer to homothetic lattices if there exists the following relation between them:

for integers , , , and satisfying the identity

.

We can “encode” this information into a matrix (see Matrices) which is an element of the group (see Groups) called . It is the group of matrices with integer entries and determinant equal to . Actually, the matrix with entries , , , and and the matrix with entries , , , and specify the same transformation, therefore what we actually want is the group called , also known as the modular group, and also written , obtained from the group by considering two matrices to be equivalent if one is the negative of the other.

We now have the moduli space that we want – we start with the upper half plane , and then we identify two points if we can map one point into the other via the action of an element of the modular group, as we have discussed earlier. In technical language, we say that they belong to the same orbit. We can write our moduli space as (the notation means that the group acts on “on the left”).

When dealing with quotient sets, which are sets of equivalence classes, we have seen in Modular Arithmetic and Quotient Sets that we can choose from an equivalence class one element to serve as the “representative” of this equivalence class. For our moduli space , we can choose for the representative of an equivalence class a point from the “fundamental domain” for the modular group. Any point on the upper half plane can be obtained by acting on a point from the fundamental domain with an element of the modular group. The following diagram, courtesy of user Fropuff on Wikipedia, shows the fundamental domain in gray:

The other parts of the diagram show where the fundamental domain gets mapped to by certain special elements, in particular the “generators” of the modular group, which are the two elements where , , , and , and , , , and . We will not discuss too much of these concepts for now. Instead we will give a preview of some concepts related to this moduli space. Topologically, this moduli space looks like a sphere with a missing point; in order to make the moduli space into a sphere (topologically), we take the union of the upper half plane with the projective line (see Projective Geometry) . This projective line may be thought of as the set of all rational numbers together with a “point at infinity.” The modular group also acts on this projective line, so we can now take the quotient of (denoted by the same equivalence relation as earlier; this new space, topologically equivalent to the sphere, is called the modular curve.

The functions and “differential forms” on the modular curve are of special interest. They can be obtained from functions on the upper half plane (with the “point at infinity”) satisfying certain conditions related to the modular group. These functions are called weakly modular functions; if they are holomorphic everywhere except at the “point at infinity”, they are called modular functions and if they are holomorphic everywhere, including the “point at infinity”, they are called modular forms. Modular forms are an interesting object of study in themselves, and their generalizations, automorphic forms, are a very active part of modern mathematical research.

An elliptic curve (not to be confused with an ellipse) is a certain kind of polynomial equation which can usually be expressed in the form

where and satisfy the condition that the quantity

is not equal to zero. This is not the most general form of an elliptic curve, as it will not hold for coefficients of “finite characteristic” equal to or ; however, for our present purposes, this definition will suffice.

Examples of elliptic curves are the following:

which, for real and may be graphed in the “Cartesian” or “–” plane as follows (image courtesy of user YassineMrabet of Wikipedia):

This rather simple mathematical object has very interesting properties which make it a central object of study in many areas of modern mathematical research.

In this post we focus mainly on one of these many interesting properties, which is the following:

The points of an elliptic curve form a group.

A group is a set with a law of composition which is associative, and the set contains an “identity element” under this law of composition, and every element of this set has an “inverse” (see Groups). Now this law of composition applies whether the points of the elliptic curve have rational numbers, real numbers, or complex numbers for coordinates, and it is always given by the same formula. It is perhaps most visible if we consider real numbers, since in that case we can plot it on the – plane as we have done earlier. The law of composition is also often called the “tangent and chord” or “tangent and secant” construction.

We now expound on this construction. Given two points on the elliptic curve and on the curve, we draw a line passing through both of them. In most cases, this line will pass through another point on the curve. Then we draw a vertical line that passes through the point . This vertical line will pass through another point on the curve. This gives us the law of composition of the points of the elliptic curve, and we write . Here is an image courtesy of user SuperManu of Wikipedia:

The usual case that we have described is on the left; the other three images show other different cases where the line drawn does not necessarily go through three points. This happens, for example, when the line is tangent to the curve at some point , as in the second picture; in this case, we think of the line as passing through twice. Therefore, when we compute , the third point is itself, and it is through that we draw our vertical line to locate , which is equal to .

The second picture also shows another computation, that of , or . Again, since this necessitates taking a line that passes through the point twice, this means that the line must be tangent to the elliptic curve at . The third point that it passes through is the point , and we draw the vertical line through to find the point , which is equal to .

Now we discuss the case described by the third picture, where the line going through the two points and which we want to “add” is a vertical line. To explain what happens, we need the notion of a “point at infinity” (see Projective Geometry). We write the point at infinity as , expressing the idea that it is the identity element of our group. We cannot find this point at infinity in the – plane, but we can think of it as the third point that the vertical line passes through aside from and . In this case, of course, there is no need to draw another vertical line – we simply write .

Finally we come to the case described by the fourth picture; this is simply a combination of the earlier cases we have described above. The vertical line is tangent to the curve at the point , so we can think of it as passing through twice, and the third point is passes through is the point at infinity , so we can write .

We will not prove explicitly that the points form a group under this law of composition, i.e. that the conditions for a set to form a group are satisfied by our procedure, but it is an interesting exercise to attempt to do so; readers may try it out for themselves or consult the references provided at the end of the post. It is worth mentioning that our group is also an abelian group, i.e. we have , and hence we have written our law of composition “additively”.

Now, to make the group law apply even when and are not real numbers, we need to write this procedure algebraically. This is a very powerful approach, since this allows us to operate with mathematical concepts even when we cannot visualize them.

Let and be the and coordinates of a point , and let and be the and coordinates of another point . Let

be the slope of the line that connects the points and . Then the point has and coordinates given by the following formulas:

In the case that is the same point as , then we define the slope of the tangent line to the elliptic curve at the point using the formula

where is the coefficient of in the formula, of the elliptic curve, i.e.

.

Then the and coordinates of the point are given by the same formulas as above, appropriately modified to reflect the fact that now the points and are the same:

This covers the first two cases in the image above; for the third case, when and are distinct points and , we simply set . For the fourth case, when and refer to the same point, and , we set . The point at infinity itself can be treated as a mere point and play into our computations, by setting , reflecting its role as the identity element of the group.

The group structure on the points of elliptic curves have practical applications in cryptography, which is the study of “encrypting” information so that it cannot be deciphered by parties other than the intended recipients, for example in military applications, or when performing financial transactions over the internet.

On the purely mathematical side, the study of the group structure is currently a very active field of research. An important theorem called Mordell’s theorem states that even though there may be an infinite number of points whose coordinates are given by rational numbers (called rational points), these points may all be obtained by performing the “tangent and chord” or “tangent and secant” construction on a finite number of points. In more technical terms, the group of rational points on an elliptic curve is finitely generated.

There is a theorem concerning finitely generated abelian groups stating that any finitely generated abelian group is isomorphic to the direct sum of copies of the integers and a finite abelian group called the torsion subgroup of . The number is called the rank of . The famous Birch and Swinnerton-Dyer conjecture, which currently carries a million dollar prize for its proof (or disproof), concerns the rank of the finitely generated abelian group of rational points on an elliptic curve.

Another thing that we can do with elliptic curves is use them to obtain representations of Galois groups (see Galois Groups). A representation of a group on a vector space over a field is a homomorphism from to , the group of bijective linear transformations of the vector space to itself. We know of course from Matrices that linear transformations of vector spaces can always be written as matrices (in our case the matrices must have nonzero determinant to ensure that the linear transformations are bijective). Representation theory allows us to study the objects of abstract algebra using the methods of linear algebra.

To any elliptic curve we can associate a certain algebraic number field (see Algebraic Numbers). The elements of these algebraic number fields are “generated” by the algebraic numbers that provide the coordinates of “-torsion” points of the elliptic curve, i.e. those points for which for some prime number .

The set of -torsion points of the elliptic curve is a -dimensional vector space over the finite field (see Modular Arithmetic and Quotient Sets), also written as . Among other things this means that we can choose two -torsion points and of the elliptic curve such that any other -torsion point can be written as for integers and between and . When an element of the Galois group of the algebraic number field generated by the coordinates of the -torsion points of the elliptic curve permutes the elements of the algebraic number field, it also permutes the -torsion points of the elliptic curve. This permutation can then be represented by a matrix with coefficients in .

The connection between Galois groups and elliptic curves is a concept that is central to many developments and open problems in mathematics. It plays a part, for example in the proof of the famous problem called Fermat’s Last Theorem. It is also related to the open problem called the Kronecker Jugendtraum (which is German for Kronecker’s Childhood Dream, and named after the mathematician Leopold Kronecker), also known as Hilbert’s Twelfth Problem, which seeks a procedure for obtaining all field extensions of algebraic number fields whose Galois group is an abelian group. This problem has been solved only in the special case of imaginary quadratic fields, and the solution involves special kinds of “symmetries” of elliptic curves called complex multiplication (not to be confused with the multiplication of complex numbers). David Hilbert, who is one of the most revered mathematicians in history, is said to have referred to the theory of complex multiplication as “…not only the most beautiful part of mathematics but of all science.”